Polynomial formulas for signed pancake stacks requiring five to nine flips

Let $R_k^B(n)$ denote the number of signed permutations of size $n$ whose burnt-pancake distance from the sorted stack is $k$. Polynomial formulas conjecture. If $n\geq1$, then

$$R^B_5(n)=\frac{1}{6}n(n-1)(n-2)(6n^2-17n+3),$$ $$R^B_6(n)=\frac{1}{60}n(n-1)(n-2)(60n^3-343n^2+401n+284),$$ $$R^B_7(n)=\frac{1}{240}n(n-1)(n-2)(n-3)(240n^3-1499n^2+925n+5104),$$ $$R^B_8(n)=\frac{1}{5040}n(n-1)(n-2)(n-3)(5040n^4-52123n^3+113415n^2+314716n-1027242),$$

and

$$R^B_9(n)=\frac{1}{40320}(n-1)(n-2)(n-3)(n-4)(40320n^5-444061n^4+644746n^3+6638777n^2-18991470n).$$

These formulas were obtained by polynomial fitting and also explain the zero entries in the cited data table; the source provides no proof or resolution, so the conjecture remains open.